02141493v53n1p73

Publ. Mat. 53 (2009), 73–82
ON FIXED POINTS OF AUTOMORPHISMS OF
NON-ORIENTABLE UNBORDERED KLEIN SURFACES
Grzegorz Gromadzki
Abstract
In 1973, Macbeath found a general formula for the number of
points fixed by an arbitrary orientation preserving automorphism
of a Riemann surface X. It was given in terms of a group Gof con-
formal automorphisms of Xand the ramification data of the cover-
ing X→X/G, which corresponds to the so called universal cover-
ing transformation group. In these terms, for the case of a cyclic
group of automorphisms of an unbordered non-orientable Klein
surface, the formula was given later by Izquierdo and Singerman
and here we find formulas valid for an arbitrary (finite) group G
of automorphisms.
1. Introduction
It is not difficult to see, that the set of fixed points of an automor-
phism of a compact Riemann surface consists either of isolated points
or simple closed Jordan curves called ovals (and sometimes mirrors).
The last case occurs only for anticonformal involutions, in the literature
known as symmetries. In 1973, Macbeath found in [14] a formula for
the number of points fixed by an arbitrary automorphism of a Riemann
surface Xin terms of the group of conformal automorphisms of Xand
the ramification data of its action, which corresponds to the so called
universal covering transformation group. Later we found (in [7] —see
also [8]) a similar formula for the number of ovals of a symmetry.
It is worth mentioning here that the possible number of ovals of a sym-
metry of a Riemann surface is given by the classical Harnack-Weichold
Theorem [10], [17]. A useful role is also played by the method of Hoare-
Singerman [11], concerning non-normal subgroups of NEC-groups, which
2000 Mathematics Subject Classification. Primary: 30F; Secondary: 14H.
Key words. Automorphisms of Riemann and Klein surfaces, fixed-point set, Fuchsian
and NEC-groups, uniformization.
Supported by SAB2005-0049 and by MTM2005-01637 of the Spanish Ministry of
Sciences and Education.

74 G. Gromadzki
allowed the relations between the total number of ovals of two symme-
tries and the order of their product to be found in [4], and the nature
of the set of fixed points of involutions of compact Klein surfaces with
boundary in [3].
The case of non-orientable surfaces is essentially different, since both
types of fixed points may occur simultaneously. For a cyclic group G
of automorphisms, Izquierdo and Singerman found [12] formulas for the
number of isolated fixed points and the number of ovals of any automor-
phisms from G, in terms of the universal covering transformation group.
Here we find such formulas for an arbitrary group of automorphisms of
such a surface. We also give some illustrative examples.
2. Preliminaries
We shall use combinatorial methods based on the Riemann uni-
formization theorem and theory of Fuchsian and NEC-groups as in [5],
where the reader can find necessary concepts and facts together with
the precise references to the original sources. By an unbordered non-
orientable Klein surface we mean a non-orientable compact topological
surface with a dianalytic structure which, roughly speaking, differs from
the classical analytic one by the fact, that the complex conjugation is
allowed for transition functions of charts, see [1] for preciseness. The
principal role in combinatorial study of such surfaces is nowadays being
played by the counterpart of the Riemann uniformization theorem, by
which such a surface Xcan be represented as the orbit space H/Γ of the
hyperbolic plane, with respect to the action of some, so called surface
NEC-group [1]. The notion of a Klein surface has already been known to
Klein himself, but until seventies such surfaces have been studied either
as algebraic curves with real equations or as Riemann surfaces together
with a single antiholomorphic involution by considering automorphisms
commuting with it. By [1], for a surface given as such orbit space, its
group of automorphisms Gcan be represented as the factor group Λ/Γ
for some other NEC-group Λ and the pair (H,Λ) is called universal cov-
ering transformation group for the action (G, X).
An NEC-group is a discrete subgroup of the group of isometries G
of the hyperbolic plane H, including those reversing orientation with a
compact orbit space. If Λ contains no orientation preserving elements of
finite order, then it is called surface NEC-group. Using fundamental re-
gions, Macbeath and Wilkie [13], [18] associated to every NEC-group Λ
the so called signature, which determines its algebraic structure. It has

Fixed Points of Automorphisms of Klein Surfaces 75
the form
(1) (g;±; [m1,...,mr]; {(n11,...,n1s1),...,(nk1,...,nksk)})
and mentioned above surface groups have signatures (g;±; [−]; {−}).
The numbers mi≥2 are called the proper periods, the brackets
(ni1,...,nisi), the period cycles, the numbers nij ≥2 are the link peri-
ods and g≥0 is said to be the orbit genus of Λ. The orbit space H/Λ
is a surface with kboundary components, orientable or not according
to the sign being + or −, having topological genus gand the canonical
projection H → H/Λ is a covering ramified over rinterior points with
ramification indices miand over sipoints lying on each boundary com-
ponent with ramification indices nij . A group with the signature (1) has
a presentation given by generators:
(a) xi,i= 1,...,r, (elliptic elements)
(b) cij ,i= 1, . . . , k, j = 0, . . . , si,(hyperbolic reflections)
(c) ei,i= 1,...,k, (boundary generators)
(d) ai, bi,i= 1,...,g if the sign is +,(hyperbolic translations)
di,i= 1,...,g if the sign is −,(glide reflections)
and relations
(1) xmi
i= 1, i=1,...,r,
(2) cisi=e−1
ici0ei,i=1,...,k,
(3) c2
ij = 1, i=1, . . . , k, j = 0,...,si,
(cij−1cij )nij = 1, i=1, . . . , k, j = 1,...,si,
(4) x1...xre1...eka1b1a
−1
1b
−1
1. . . agbga
−1
gb
−1
g=1 or
x1...xre1. . . ekd2
i...d2
g= 1.
Any system of generators of an NEC-group, which satisfies the above
relations, will be called a canonical system of generators and it is known,
that every element of finite order in Λ is conjugate either to a canonical
reflection or to a power of some canonical elliptic element xior else to a
power of the product of two consecutive canonical reflections cij−1,cij .
For every NEC-group we have an associated fundamental region,
whose hyperbolic area µ(Λ) depends only on the group and for a group
with signature (1) it is given by
(2) 2π εg +k−2 +
r
X
i=1
(1 −1/mi) + 1/2
k
X
i=1
si
X
i=1
(1 −1/nij)!,

76 G. Gromadzki
where ε= 2 or 1 according to the sign being + or −. It is known that an
abstract group with the presentation given by the above generators and
relations can be realized as an NEC-group with the signature (1) if and
only if (2) is positive. Finally, if Λ′is a subgroup of finite index in an
NEC-group Λ, then it is an NEC-group itself and the Hurwitz-Riemann
formula (known also as the Hurwitz formula) says that
(3) [Λ : Λ′] = µ(Λ′)/µ(Λ).
3. On fixed points of an automorphism of unbordered
non-orientable Klein surfaces
Let X=H/Γ be a non-orientable, unbordered Klein surface, where
Γ is an NEC-group with signature (g;−; [−]; {−}) and let the action
of Gon Xbe defined by an epimorphism θ: Λ →G, where Λ is an
NEC-group with signature (1). We start this principal section of the
paper with the lemma describing the nature of the set of points fixed by
an automorphism of X.
Lemma 3.1. The set of points fixed by an automorphism of an unbor-
dered, non-orientable Klein surface consists of isolated points and ovals.
Proof: The canonical projection π:H → Xis a non-ramified covering
and the action of Gon Xis defined by
gx =π(λh) if g=θ(λ), x =π(h).
So if ϕ=θ(λ), then x=π(h) is its fixed point if and only if λh =γh for
some γ∈Γ, which means that γ−1λfixes h. But then γ−1λis either an
elliptic element or a reflection, say c, since these are the only hyperbolic
isometries with fixed points. In the first case xis obviously an isolated
fixed point. Now, if in the second case ℓis the axis of c, then on the
one hand π(ℓ) is the fixed-point set of ϕ, while on the other hand it is
homeomorphic to a circle, as πis an unramified covering.
Theorem 3.2. The number of isolated fixed points of ϕ∈Gon Xis
given by the formula
|NG(hϕi)|X1/mi+X1/nij,
where Nstands for the normalizer and the sums are taken respectively
over canonical elliptic generators and consecutive canonical reflections
for which ϕis conjugate to a power of θ(xi)and θ(cij−1cij )respectively.

Fixed Points of Automorphisms of Klein Surfaces 77
Theorem 3.3. An involution σof Xhas
X[C(G, θ(c)) : θ(C(Λ, c))]
ovals, where Cdenotes the centralizer and the sum is taken over non-
conjugate canonical reflections of Λ, whose images under θare conjugate
to σin G.
The proof of Theorem 3.2: If ϕhas order n, then the number of its fixed
points equals the number of the conjugacy classes of cyclic subgroups of
order nof Γϕ=θ−1(hϕi). Clearly each such subgroup is generated by
an elliptic element. However, each elliptic element is conjugate either to
a power of some canonical elliptic generator or to a power of the product
of two consecutive canonical reflections of Λ and we say that fixed points
of ϕare produced by xior cij−1cij in each of these cases respectively. We
have to find how many fixed points of ϕis produced by each element xi
and by each product cij−1cij .
Assume first, that xiproduces fixed points of ϕ. Then, since conjugate
elements have the same number of fixed points, by exchanging ϕwith a
suitable conjugation, we may assume that xni
i∈Γϕfor ni=mi/n. Now
wxni
iw−1∈Γϕif and only if w∈θ−1(NG(hϕi)). Observe however, that
wxni
iw−1and w′xni
iw′−1are conjugate in Γϕif and only if w−1γw′∈
NΛ(hxni
ii) = hxiifor some γ∈Γϕ, which means that w−1w′∈ hxiiΓϕ.
Thus xiproduces
[θ−1(NG(hϕi)) : hxiiΓϕ] = [θ−1(NG(hϕi))/Γ : hxiiΓϕ/Γ] = |NG(hϕi)|/mi
fixed points of ϕ.
Similarly, assume that cij−1cij produces fixed points of ϕ. Then
again we may assume that (cij−1cij )mij ∈Γϕfor mij =nij /n. Also
w(cij−1cij )mij w−1∈Γϕif and only if w∈θ−1(NG(hϕi)). On the other
hand w(cij−1cij )mij w−1and w′(cij−1cij )mij w′−1are conjugate in Γϕ
if and only if w−1γw′belongs to NΛ(h(cij−1cij )mij i) = hcij−1cij i, for
some γ∈Γϕ, which means that w−1w′∈ hcij−1cij iΓϕ. Hence cij−1cij
produces
[θ−1(NG(hϕi)) : hcij−1cij iΓϕ] = [θ(θ−1(NG(hϕi))) : θ(hcij−1cij iΓϕ)]
= [NG(hϕi) : hθ(cij−1cij )i]
=|NG(hϕi)|/nij
fixed points of ϕ.